Let's look at the sine of the sum of 3 angles and combinations of sum and differences:
(1) sin(X+Y+Z) = sin(X+(Y+Z)) = sinXcos(Y+Z) + cosXsin(Y+Z) = sinXcosYcosZ - sinXsinYsinZ + cosXsinYcosZ + cosXcosYsinX. If we keep X, Y and Z in order we can use c and s in the right order to reduce the notation. So we have sin(X+Y+Z)=scc-sss+ccs-csc+ccs.
(2) sin(X-(Y+Z))=cosXsin(Y+Z)-sinXcos(Y+Z)=csc+ccs-scc+sss.
If we subtract (2) from (1), some terms cancel and others double up: (3) 2scc-2sss.
(4) sin(X+(Y-Z))=sinXcos(Y-Z)+cosXsin(Y-Z)=scc+sss+ccs-csc.
(5) sin(X-(Y-Z))=cosXsin(Y-Z)-sinXcos(Y-Z)=ccs-csc-scc-sss.
Subtract (5) from (4): (6) 2scc+2sss.
Add (3) and (6): (7) 4scc=4sinXcosYcosZ=4cosYcosZsinX, where we can put Y=A, Z=B, X=C.
Therefore:
4cosAcosBsinC=(3)+(6)=(1)-(2)+(4)-(5)=sin(X+Y+Z)-sin(X-Y-Z)+sin(X+Y-Z)-sin(X-Y+Z)=
sin(A+B+C)-sin(C-A-B)+sin(C+A-B)-sin(C-A+B).
We have 3 terms on the left side of the equation so one of the above terms must be zero. Let's start with C-A-B=n(pi), where n is an integer, then C=A+B+n(pi). The equation then becomes: sin(2A+2B+n(pi))+sin(2A+n(pi))-sin(2B+n(pi)=4cosAcosBsinC. So if 2A+2B+n(pi)=2A, then B=-n(pi)/2 and C=A-n(pi)/2. sin(2A+n(pi)) must equal sin2B so 2B=2A+n(pi)=-n(pi), B=-n(pi)/2=A+n(pi)/2, so A=-n(pi) and C=A-n(pi)/2=-3n(pi)/2. Yes, this fits the original equation.
Are there any more solutions? Next try A+B+C=n(pi) so C=n(pi)-(A+B) and we have: -sin(n(pi)-2A-2B)+sin(n(pi)-2B)-sin(n(pi)-2A)=sin(2A+2B-n(pi))-sin(2B-n(pi))+sin(2A-n(pi))=4cosAcosBsinC. This time 2A+2B-n(pi)=2A, so B=n(pi)/2 and C=n(pi)/2-A, 2B=n(pi)=2A-n(pi), A=3n(pi)/2 and C=n(pi)/2-A=-n(pi).
Try C+A-B=n(pi), so C=B-A+n(pi). We have: sin(2B+n(pi))-sin(n(pi)-2A)-sin(2B-2A+n(pi))=sin(2B+n(pi))+sin(2A-n(pi))-sin(2B-2A+n(pi)). So 2C=2B-2A+n(pi), C=B-A+n(pi)/2; but C=B-A+n(pi) and n(pi)/2 does not equal n(pi), and this path reveals no solution.
The solution of all zero angles is only one of many. If we put n=1 or -1 we get a sample of other solutions given here as ordered triplets, (A,B,C) in degrees:
(180,90,270), (270,90,-180). Put n=2 or -2: (360,180,540), (540,180,-360).